High-performance electrical system design requires the use of sophisticated interconnections between system components. These interconnections must be designed to achieve three interrelated objectives: minimize signaling delay between components; minimize electromagnetic cross talk noise between interconnections; and ensure immunity to external electromagnetic interferences. Two main examples of these interconnections are multiconductor transmission lines and connectors.
Transmission lines are interconnects whose electrical length is large compared with the wavelength of the transmitted signal. A transmission line macromodel is a mathematical model that enables analysis and simulation of transmission lines in electrical circuits. Passivity of the macromodel is advantageous because it results in a more robust, reliable and efficient transmission line analysis.
Multiconductor transmission lines are present throughout any electrical system comprising several integrated circuits (chips). They are used on a printed circuit board (PCB) for signal transmission between different chipsets. Multiconductor transmission lines are also used to transfer signals inside a package containing the chip as well as to transfer signals inside the chip itself. PCB and package transmission lines are known in the art as off-chip transmission lines while the lines responsible for transferring signals within the chip are known in the art as on-chip transmission lines. The transmission line nature of an interconnection depends on the wavelength of the signal carried by the interconnection. With the constant increase in electronic system speed, the signal wavelengths are becoming shorter. The net result is that more and more of the interconnections are behaving as transmission lines, which makes the task of modeling and analyzing overwhelming. The situation is rendered even more complex by the fact that the on-chip and off-chip transmission lines behave very differently in terms of the losses (i.e., attenuation and the distortion) incurred by the signals they carry. Because of their small cross sections, on-chip transmission lines are very lossy relative to off-chip transmission lines. Among off-chip transmission lines, packaging transmission lines are in general more lossy than PCB transmission lines. These differences in location (on-chip, off-chip), length (short, long), losses (high, low), and signal wavelength make the efficient modeling, simulation, and analysis of transmission lines a difficult engineering task.
In an article by A. Dounavis et al entitled “Delay extraction and passive macromodeling of lossy coupled transmission lines” in Digest of Electrical Performance of Electronic Packaging, volume 12, pages 251-254, Princeton, N.J., October 2003, a passive macromodel for lossy, dispersive multiconductor transmission lines has been proposed. The article is hereby incorporated herein by reference. The macromodel was a multiplicative approximation of the matrix exponential known as the Lie product. The circuit implementation of the macromodel is a cascade of elementary cells, each cell being the combination of a pure delay element and a lumped circuit representing the transmission line losses.
The present invention combines the results of the Dounavis et al paper with transmission line theory to derive a time-domain error criteria for the Lie product macromodel. The error criteria is then used to derive a lower bound on the number of cells needed in the macromodel so that the worst-case magnitude of the time-domain error is less than a given engineering tolerance.